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Smale-Barden manifolds are simply-connected closed 5-manifolds. It is animportant and difficult question to decide when a Smale-Barden manifold admitsa Sasakian or a K-contact structure. The known constructions of Sasakian andK-contact structures are obtained mainly by two techniques. These are eitherlinks (Boyer and Galicki), or semi-regular Seifert fibrations over smoothorbifolds (Koll\'ar). Recently, the second named author of this article startedthe systematic development of quasi-regular Seifert fibrations, that is, overorbifolds which are not necessarily smooth. The present work is devoted toseveral applications of this theory. First, we develop constructions of aSmale-Barden manifold admitting a quasi-regular Sasakian structure but not asemi-regular K-contact structure. Second, we determine all Smale-Bardenmanifolds that admit a null Sasakian structure. Finally, we show acounterexample in the realm of cyclic K\"ahler orbifolds to thealgebro-geometric conjecture that claims that for an algebraic surface with$b_1=0$ and $b_2>1$ there cannot be $b_2$ smooth disjoint complex curves ofgenus g>0 spanning the (rational) homology.

Quasi-regular Sasakian and K-contact structures on Smale-Barden manifolds